Note 1 to entry: General theory of relativity is valid in any space-time including curved space-time that interprets gravity as a metric property of space-time.

Note 2 to entry: The metric tensor cannot be reduced to the diagonal form, and thus the metric is neither Euclidean nor pseudo-Euclidean any more, and the use of complex numbers has no advantage. The scalar product of two four-vectors and the length of a displacement are not expressed simply by a Pythagorean rule. Moreover, the metric tensor is generally different for different events. Thus, the concept of position four-vector and even the finite displacement four-vector are not easily applicable in GTR. Instead, an infinitesimal displacement four-vector is to be used.